%I #5 Apr 10 2020 14:39:55
%S 1,1,1,2,1,1,6,1,1,2,2,24,1,1,1,2,2,6,120,1,1,1,2,2,1,6,6,4,24,720,1,
%T 1,1,2,1,1,6,2,2,2,24,6,12,120,5040,1,1,1,2,1,1,6,2,1,2,2,24,2,4,2,6,
%U 120,24,12,48,720,40320
%N Irregular triangle where row n lists the product of the factorials of the exponentials of the partitions of n and the partitions are enumerated in canonical order.
%C By 'canonical order' we understand the graded reverse lexicographic order (the default order of Mathematica and SageMath).
%e The irregular table starts:
%e [0] [1]
%e [1] [1]
%e [2] [1, 2]
%e [3] [1, 1, 6]
%e [4] [1, 1, 2, 2, 24]
%e [5] [1, 1, 1, 2, 2, 6, 120]
%e [6] [1, 1, 1, 2, 2, 1, 6, 6, 4, 24, 720]
%e [7] [1, 1, 1, 2, 1, 1, 6, 2, 2, 2, 24, 6, 12, 120, 5040]
%o (SageMath)
%o def A333144row(n):
%o return [product(factorial(expo) for expo in partition.to_exp()) for partition in Partitions(n)]
%o for n in (0..9): print(A333144row(n))
%Y Row sums are A161779.
%Y Cf. A069123.
%K nonn,tabf
%O 0,4
%A _Peter Luschny_, Apr 10 2020